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id="post-info"><h1 class="post-title">《机器学习》 西瓜书习题 第 6 章: 支持向量机</h1><div id="post-meta"><div class="meta-firstline"><span class="post-meta-date"><i class="far fa-calendar-alt fa-fw post-meta-icon"></i><span class="post-meta-label">发表于</span><time class="post-meta-date-created" datetime="2020-02-26T09:35:00.000Z" title="发表于 2020-02-26 17:35:00">2020-02-26</time><span class="post-meta-separator">|</span><i class="fas fa-history fa-fw post-meta-icon"></i><span class="post-meta-label">更新于</span><time class="post-meta-date-updated" datetime="2021-08-14T04:39:34.319Z" title="更新于 2021-08-14 12:39:34">2021-08-14</time></span><span class="post-meta-categories"><span class="post-meta-separator">|</span><i class="fas fa-inbox fa-fw post-meta-icon"></i><a class="post-meta-categories" href="/categories/%E6%9C%BA%E5%99%A8%E5%AD%A6%E4%B9%A0/">机器学习</a><i class="fas fa-angle-right post-meta-separator"></i><i class="fas fa-inbox fa-fw post-meta-icon"></i><a class="post-meta-categories" href="/categories/%E6%9C%BA%E5%99%A8%E5%AD%A6%E4%B9%A0/%E8%A5%BF%E7%93%9C%E4%B9%A6/">西瓜书</a><i class="fas fa-angle-right post-meta-separator"></i><i class="fas fa-inbox fa-fw post-meta-icon"></i><a class="post-meta-categories" href="/categories/%E6%9C%BA%E5%99%A8%E5%AD%A6%E4%B9%A0/%E8%A5%BF%E7%93%9C%E4%B9%A6/%E4%B9%A0%E9%A2%98/">习题</a></span></div><div class="meta-secondline"><span class="post-meta-separator">|</span><span class="post-meta-pv-cv"><i class="far fa-eye fa-fw post-meta-icon"></i><span class="post-meta-label">阅读量:</span><span id="busuanzi_value_page_pv"></span></span><span class="post-meta-separator">|</span><span class="post-meta-commentcount"><i class="far fa-comments fa-fw post-meta-icon"></i><span class="post-meta-label">评论数:</span><a href="/ML/watermelon_book/answer/6/#post-comment"><span id="twikoo-count"></span></a></span></div></div></div></header><main class="layout" id="content-inner"><div id="post"><article class="post-content" id="article-container"><h1 id="习题"><a href="#习题" class="headerlink" title="习题"></a>习题</h1><h2 id="6-1"><a href="#6-1" class="headerlink" title="6.1"></a>6.1</h2><p><strong>试证明样本空间中任意点 $\boldsymbol{x}$ 到超平面 $(\boldsymbol{w}, b)$ 的距离为式 $(6.2)$ .</strong></p>
<p><img src="1.png"></p>
<p>设超平面为 $\ell(\boldsymbol{w}, b)$ , $\boldsymbol{x}$ 在 $\ell$ 上的投影为 $\boldsymbol{x_0}$ , 离超平面的距离为 $r$ . 容易得<br>$$<br>\boldsymbol{w}\perp \ell\\\boldsymbol{x} = \boldsymbol{x_0} + r\frac{\boldsymbol{w}}{||\boldsymbol{w}||}\\\boldsymbol{w}^\mathrm{T}\boldsymbol{x_0} + b = 0<br>$$<br>则有<br>$$<br>\boldsymbol{x_0} = \boldsymbol{x} - r\frac{\boldsymbol{w}}{||\boldsymbol{w}||}<br>$$</p>
<p>$$<br>\begin{aligned}<br>\boldsymbol{w}^\mathrm{T}(\boldsymbol{x} - r\frac{\boldsymbol{w}}{||\boldsymbol{w}||}) + b &amp;= 0\\<br>\boldsymbol{w}^\mathrm{T}\boldsymbol{x} - r\frac{\boldsymbol{w}^\mathrm{T}\boldsymbol{w}}{||\boldsymbol{w}||} + b &amp;= 0\\<br>\boldsymbol{w}^\mathrm{T}\boldsymbol{x} - r\frac{||\boldsymbol{w}||^2}{||\boldsymbol{w}||} + b &amp;= 0\\<br>\boldsymbol{w}^\mathrm{T}\boldsymbol{x} + b &amp;= r\frac{||\boldsymbol{w}||^2}{||\boldsymbol{w}||}\\<br>\end{aligned}<br>$$</p>
<p>即得<br>$$<br>r = \frac{\boldsymbol{w}^\mathrm{T}\boldsymbol{x} + b}{||\boldsymbol{w}||}<br>$$<br>由于距离是大于等于 $0$ 的, 所以结果再加上绝对值<br>$$<br>r = \frac{\left|\boldsymbol{w}^\mathrm{T}\boldsymbol{x} + b\right|}{||\boldsymbol{w}||}\tag{6.2}<br>$$</p>
<h2 id="6-2"><a href="#6-2" class="headerlink" title="6.2"></a>6.2</h2><p><strong>试使用 $\mathrm{LIBSVM}$ , 在西瓜数据集 $3.0\alpha$ 上分别用线性核和高斯核训练一个 $\mathrm{SVM}$ , 并比较其支持向量的差别.</strong></p>
<p><a target="_blank" rel="noopener" href="https://aistudio.baidu.com/aistudio/projectdetail/259787">《机器学习》西瓜书 第 6 章 编程实例</a></p>
<h2 id="6-3"><a href="#6-3" class="headerlink" title="6.3"></a>6.3</h2><p><strong>选择两个 $\mathrm{UCI}$ 数据集, 分别用线性核和高斯核训练一个 $\mathrm{SVM}$ , 并与 $\mathrm{BP}$ 神经网络和 $\mathrm{C4.5}$ 决策树进行实验比较.</strong></p>
<p><a target="_blank" rel="noopener" href="https://aistudio.baidu.com/aistudio/projectdetail/259787">《机器学习》西瓜书 第 6 章 编程实例</a></p>
<h2 id="6-4"><a href="#6-4" class="headerlink" title="6.4"></a>6.4</h2><p><strong>试讨论线性判别分析与线性核支持向量机在何种条件下等价.</strong></p>
<p>线性判别分析能够解决 $n$ 分类问题, 而 $\mathrm{SVM}$ 只能解决二分类问题, 如果要解决 $n$ 分类问题要通过 $\mathrm{OvR(One\ vs\ Rest)}$ 来迂回解决.</p>
<p>线性判别分析能将数据以同类样例间低方差和不同样例中心之间大间隔来投射到一条直线上, 但是如果样本线性不可分, 那么线性判别分析就不能有效进行, 支持向量机也是.<br>综上, 等价的条件是:</p>
<ul>
<li>数据有且仅有 2 种, 也就是说问题是二分类问题.</li>
<li>数据是线性可分的.</li>
</ul>
<h2 id="6-5"><a href="#6-5" class="headerlink" title="6.5"></a>6.5</h2><p><strong>试述高斯核 $\mathrm{SVM}$ 与 $\mathrm{RBF}$ 神经网络之间的联系.</strong></p>
<p>实际上都利用了核技巧, 将原来的数据映射到一个更高维的空间使其变得线性可分.</p>
<h2 id="6-6"><a href="#6-6" class="headerlink" title="6.6"></a>6.6</h2><p><strong>试析 $\mathrm{SVM}$ 对噪声敏感的原因.</strong></p>
<p>&nbsp;$\mathrm{SVM}$ 的特性就是 “支持向量” . 即线性超平面只由少数 “支持向量” 所决定. 若噪声成为了某个 “支持向量” —— 这是非常有可能的. 那么对整个分类的影响是巨大的. 反观对率回归, 其线性超平面由所有数据共同决定, 因此一点噪声并无法对决策平面造成太大影响.</p>
<h2 id="6-7"><a href="#6-7" class="headerlink" title="6.7"></a>6.7</h2><p><strong>试给出试 $(6,52)$ 的完整 $\mathrm{KKT}$ 条件.</strong></p>
<p>&nbsp;$\mathrm{KKT}$ 条件:<br>$$<br>\begin{cases}<br>\xi_i \geqslant 0\\<br>\hat{\xi}_i \geqslant 0\\<br>f(\boldsymbol{x}_i) - y_i - \epsilon - \xi_i \leqslant 0\\<br>y_i - f(\boldsymbol{x}_i) - y_i - \epsilon - \hat{\xi}_i \leqslant 0\\<br>\mu_i \geqslant 0\\<br>\hat{\mu}_i \geqslant 0\\<br>\alpha_i \geqslant 0\\<br>\hat{\alpha}_i \geqslant 0\\<br>\mu_i\xi_i = 0\\<br>\hat{\mu}_i\hat{\xi}_i = 0\\<br>\alpha_i(f(\boldsymbol{x}_i) - y_i - \epsilon - \xi_i) = 0\\<br>\hat{\alpha}_i(y_i - f(\boldsymbol{x}_i) - y_i - \epsilon - \hat{\xi}_i) = 0<br>\end{cases}<br>$$</p>
<h2 id="6-8"><a href="#6-8" class="headerlink" title="6.8"></a>6.8</h2><p><strong>以西瓜数据集 $3.0\alpha$ 的 “密度” 为输入, “含糖率” 为输出, 试使用 $\mathrm{LIBSVM}$ 训练一个 $\mathrm{SVR}$.</strong></p>
<p><a target="_blank" rel="noopener" href="https://aistudio.baidu.com/aistudio/projectdetail/259787">《机器学习》西瓜书 第 6 章 编程实例</a></p>
<h2 id="6-9"><a href="#6-9" class="headerlink" title="6.9"></a>6.9</h2><p><strong>试使用核技巧推广对率回归, 产生 “核对率回归” .</strong></p>
<blockquote>
<p>可以发现, 如果使用対率损失函数 $\ell_{log}$ 来代替式 $(6.29)$ 中的 $0/1$ 损失函数, 则几乎就得到了対率回归模型 $(3.27)$ .</p>
</blockquote>
<p>我们根据原文, 将损失函数换成 $\ell_{log}$, 再使用核技巧, 就能实现 “核対率回归” .</p>
<h2 id="6-10"><a href="#6-10" class="headerlink" title="6.10*"></a>6.10*</h2><p><strong>试设计一个能显著减少 $\mathrm{SVM}$ 中支持向量的数目而不显著降低泛化性能的方法.</strong></p>
<p>可以将一些冗余的支持向量去除到只剩必要的支持向量. 比如在二维平面, 只需要 $3$ 个支持向量就可以表达一个支持向量机, 所以我们将支持向量去除到只剩 $3$ 个.</p>
<p>更广泛的情况是, 若是 $n$ 维平面, 那么只需要 $n + 1$ 个支持向量就能表达一个支持向量机.</p>
<h1 id="资料推荐"><a href="#资料推荐" class="headerlink" title="资料推荐"></a>资料推荐</h1><ul>
<li><a target="_blank" rel="noopener" href="https://blog.csdn.net/v_july_v/article/details/7624837">支持向量机通俗导论（理解SVM的三层境界）</a></li>
</ul>
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